Below is a clear, comprehensive explanation tailored to your cortical-mesh context and consistent with how eigenmodes are treated in FEM, Laplace–Beltrami analysis, and graph signal processing.

1. What exactly is an eigenmode on a cortical mesh?

For a cortical surface discretized as a triangular FEM mesh, the Laplace–Beltrami operator is discretized as a sparse matrix:

where:

• $W$ is the stiffness matrix (cotangent Laplacian),

• $M$ is the mass matrix, and

• $(\lambda, \phi)$ is an eigenvalue/eigenfunction pair.

The eigenfunction $\phi_k$ assigns a scalar value to each vertex of the mesh.
So visually, each eigenfunction is simply a scalar field on the surface.

2. Can eigenmodes be visualized?

Yes. Each eigenfunction $\phi_k$ can be plotted by:

• mapping $\phi_k(v)$ to color intensity at vertex $v$,

• interpolating colors across triangles.

This yields a smooth spatial pattern—the analog of sine/cosine waves but adapted to the curved cortical geometry.

Important:
Even though eigenvalues/eigenfunctions are often plotted as axes in the spectral domain (the λ-axis), they are not just axes. Each eigenfunction is a spatial pattern that you can see on the cortex.

What does an eigenmode look like?

Eigenmodes resemble spherical harmonics—except they conform exactly to the cortical geometry. For example:

• The 0th eigenfunction is constant over the surface.

• Low-λ modes vary slowly, with a few broad lobes.

• Higher-λ modes oscillate rapidly across the mesh, producing fine-grained patterns.

They are the FEM generalization of Fourier modes.

3. Why each mesh has distinct eigenmodes

Eigenmodes depend entirely on the geometry and topology of the mesh.

Two different meshes—even with the same number of vertices—will have different:

• curvature,

• sulcal geometry,

• connectivity (cotangent weights),

and therefore different spatial patterns.

Thus, eigenfunctions are geometric fingerprints of your cortical mesh.

4. Are eigenvalues “axes” or “frequencies”?

Eigenvalues $\lambda_k$:

• are scalars measuring intrinsic spatial frequency,

• define a spectral axis on which filters operate,

• correspond to the “wavenumber” of each eigenmode.

But the eigenvalues are not visual—they are just numbers.
Their associated eigenfunctions are the visualizable spatial fields.

5. How to visualize an eigenfunction on the cortex

Typical workflow in your FEM-based bct system:

matlab
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[phi, lambda] = eigs(W, M, k, 'smallestabs');
mode = phi(:,i);   % i-th eigenfunction

trisurf(Faces, Vertices(:,1), Vertices(:,2), Vertices(:,3), mode);
shading interp; axis equal; colormap jet; colorbar;

This produces a map showing positive/negative lobes, gradients, oscillations, nodal lines, etc.

6. Why eigenfunctions form a basis for spectral analysis

Because eigenfunctions of the Laplace–Beltrami operator are:

• orthonormal with respect to the mass matrix:
  $\phi_i^T M \phi_j = \delta_{ij}$,

• complete: they span all FEM functions on the mesh,

• arranged by smoothness: low λ = smooth; high λ = oscillatory.

Therefore, any cortical signal $f$ can be expanded as:

This is why eigenfunctions serve as the spectral basis, even though they are also spatial patterns you can visualize.

7. How this relates to spherical harmonics

Spherical harmonics are:

• eigenfunctions of the Laplace–Beltrami operator on the sphere,

• orthonormal basis functions,

• perfectly smooth and symmetric.

FEM eigenfunctions are:

• eigenfunctions of the Laplace–Beltrami on an arbitrary mesh,

• adapted to sulci/gyri,

• not analytic; numerical.

Both allow signals to be represented in the spectral domain and visualized.

8. Summary

Eigenfunctions are not just axes.
They are geometric wave patterns that exist directly on your cortical mesh and can be rendered like any other scalar field.

If you want, I can:

1. Produce MATLAB code that visualizes dozens of eigenmodes in a grid.

2. Show how to animate wave packets by superposing neighboring eigenmodes.

3. Explain how eigenfunctions relate to traveling waves, phase gradients, and your joint (λ–ω) filters in bct.

Yes — but with an important clarification:

Eigenfunctions of the Laplace–Beltrami operator always contain spatial gradients, lobes, and nodal lines intrinsically.
These structures are not added by your plotting routine; they are mathematically encoded in the eigenfunction itself.

Here is why.

1. Why eigenfunctions automatically have these spatial structures

An eigenfunction $\phi_k$ satisfies:

This PDE forces the eigenfunction to behave like a generalized standing wave on the cortex.

This means:

1. Gradients arise because the function must vary across the manifold.

Unless $\lambda_k = 0$, the solution cannot be constant.
The differential equation forces spatial variation → hence gradients.

2. Lobes arise because eigenfunctions must change sign.

For $k \ge 2$, the solution must oscillate.
Regions where $\phi>0$ and $\phi<0$ form lobes.

3. Nodal lines (zero-crossings curves) emerge because oscillatory solutions must cross zero.

Higher eigenmodes have more oscillations → more zero-crossing curves → more complex nodal sets.

4. Oscillatory patterns increase with eigenvalue.

Large $\lambda_k$ → high spatial frequency → tightly packed oscillations.

None of this comes from the plotting routine.
It is a mathematical property of eigenfunctions of the Laplacian.

2. Why this holds even on a complicated surface like cortex

The Laplace–Beltrami operator respects the geometry of the cortical manifold.

So the eigenfunctions:

• bend through sulci,

• stretch over gyri,

• follow geodesics of the surface.

This is why each cortical mesh has its own unique eigenmodes—geometric fingerprints driven entirely by biological shape.

Even irregular curvature imposes oscillatory behavior.
There is no way to solve the LB eigenproblem without getting such structured patterns.

3. What the visualization does (and does not do)

The visualization step only:

• takes the vertexwise scalar values $\phi_k(v)$,

• maps them to colors,

• interpolates over triangles.

It does not generate the oscillations or nodal lines.
It simply reveals them.

4. Analogies (to make this intuitive)

Analogy 1 — Guitar string

Eigenfunctions = standing waves.
Plotting them doesn’t create nodes; those nodes are intrinsic to the mode shape.

Analogy 2 — Spherical harmonics

When you plot them, the banded structure was already present in the math.

Analogy 3 — Fourier modes on a line

Plotting sin(kx) doesn’t create oscillations; it exposes them.

Eigenfunctions on a mesh = the generalization of all these.

5. Therefore:

When you compute FEM eigenfunctions on your cortex mesh, every eigenmode already contains:

• smoothness constraints,

• oscillatory patterns,

• nodal lines,

• alternating sign regions,

• intrinsic spatial gradients.

Plotting merely displays what is already present.

If you want, I can:

1. Plot the first 20 eigenmodes on a standard mesh to show how the structures evolve.

2. Explain the mathematical guarantees about nodal domains (Courant nodal domain theorem).

3. Show how these eigenfunctions form the basis for your wave packet theory and joint spectral filters.

Would you like a deeper dive?

Below is a precise, mathematically grounded explanation of lobes, nodal lines, and how they relate—and do not relate—to positive/negative frequencies in the Fourier transform. This is essential conceptual foundation for mesh-based spectral analysis.

1. What are lobes in an eigenfunction?

On a surface (e.g., cortex), an eigenfunction $\phi_k(v)$ assigns a scalar value to each vertex.
Regions where $\phi_k$ is positive or negative form broad, contiguous patches.

These patches are called lobes.

• A positive lobe is a spatial region where $\phi_k > 0$.

• A negative lobe is a region where $\phi_k < 0$.

Think of lobes as the surface-analog of the crests and troughs of a vibrating membrane.

The 0th eigenfunction has no lobes (it is constant).
The 1st non-trivial eigenfunctions typically split the cortex into two large lobes.
Higher modes have many alternating lobes.

2. What are nodal lines?

A nodal line is the geometric curve on the surface where the eigenfunction equals zero:

These curves separate the positive lobes from the negative lobes.

• On a vibrating drum, nodal lines are the “silent” curves on the membrane.

• On the cortex, nodal lines weave around sulci and gyri, dividing oscillatory regions.

In 3D domains they would be nodal surfaces; on curves, nodal points.

So:

Lobes = regions of constant sign

Nodal lines = boundaries between lobes

3. Why do eigenfunctions have lobes and nodal lines?

Because they satisfy a Laplace–Beltrami eigenproblem:

This differential equation forces spatial oscillations.
The oscillations must cross zero, hence nodal lines.
Higher eigenvalues → more oscillations → more nodal lines.

This is guaranteed mathematically (Courant’s nodal domain theorem).

4. How do lobes compare to positive/negative frequencies in a Fourier transform?

They do NOT correspond.

Here’s the clear distinction.

4.1 In the Fourier transform (1D/2D):

• The axis of real numbers $\omega$ contains positive and negative frequencies.

• Positive vs. negative frequency encodes direction of oscillation in time or space.

• The signal itself is typically complex-valued; oscillations rotate clockwise or counterclockwise in the complex plane.

In Fourier:

• Negative frequency ≈ complex conjugate symmetry

• Positive/negative frequencies are about temporal or spatial direction, not about amplitude sign.

The signal in the time/space domain may be positive or negative, but the frequency axis (ω) also has positive/negative entries, which correspond to directionality of complex exponentials.

4.2 In Laplace–Beltrami eigenfunctions on a cortical mesh:

• There is no negative λ. All eigenvalues satisfy $\lambda_k \ge 0$.

• Therefore, no negative frequencies appear in the spatial spectral domain.

• The eigenfunctions live purely in real space, and are real-valued.

The sign of an eigenfunction at a vertex (positive/negative lobe) has only one meaning:

• It indicates one side of the oscillatory pattern vs. the opposite side.

Crucially:

Positive/negative lobes in eigenfunctions do NOT correspond to positive/negative frequencies in Fourier analysis.

They are merely the spatial phases of a standing wave.

5. Why the mismatch? Because the Laplacian spectrum is fundamentally different.

Fourier transform:

• Built from complex exponentials $e^{i \omega x}$

• Frequency axis contains positive and negative frequencies

• Oscillations have direction (e.g., waves traveling right vs. left)

• Negative frequency ⟷ opposite rotation direction in the complex plane

Laplace–Beltrami eigenbasis (FEM mesh):

• Built from real-valued solutions

• Eigenvalues λ are all non-negative

• Modes have spatial patterns, not directions

• Lobes reflect standing-wave structure, not propagation

Your cortical eigenmodes are not traveling-wave solutions; they are standing-wave patterns intrinsic to the geometry.

6. Where does directionality come from on a mesh?

Not from eigenfunctions, but from:

• temporal dynamics,

• complex analytic continuation (Hilbert transform),

• phase gradients over time,

• joint $(\lambda,\omega)$ tilted filters (your wave packet kernels),

• physical PDEs with time derivatives.

To detect traveling waves, you need time-varying phase, not static lobes.

7. Summary table

8. Key takeaway

Lobes and nodal lines represent spatial oscillatory structure in the eigenfunction.
They are NOT related to positive/negative frequencies of Fourier analysis.

Eigenfunctions = real standing waves on a curved surface.
Fourier modes = complex oscillations with directional meaning.

If you'd like, I can:

• Draw a visual analogy showing how nodal lines = “zero contours” similar to drum modes.

• Show mathematically why Laplacian eigenmodes cannot carry propagation direction.

• Connect this to why your wave packet filters need joint λ–ω tilting to introduce direction.

Would you like that?

Short answer: Eigenfunctions ARE generalizations of sinusoids, but they are not literally sinusoids unless the domain is flat and regular.
And yes, the FFT corresponds to eigenfunctions of the 1-D Laplacian on a periodic time domain.

Now let’s unpack this clearly and precisely.

1. Are eigenfunctions “just sinusoids”?

1.1 If the domain is a line or circle → Yes

On simple domains:

• 1D interval with Dirichlet BC → eigenfunctions are sines.

• 1D periodic domain (circle) → eigenfunctions are complex exponentials $e^{i\omega t}$, i.e., sinusoids.

• 2D infinite plane → eigenfunctions are plane waves $e^{i(k_x x + k_y y)}$.

So for standard Fourier analysis, the eigenfunctions of the Laplacian are exactly sinusoids.

This is why Fourier basis = Laplacian eigenbasis on flat domains.

1.2 On a general curved surface (like the cortex) → Not sinusoids, but sinusoid-like

They satisfy the same PDE:

but because the geometry and boundaries are complicated, the solutions:

• bend according to curvature,

• respect sulci/gyri,

• warp according to irregular mesh connectivity.

Thus:
Eigenfunctions on a curved manifold are “generalized sinusoids”: oscillatory solutions whose oscillation shape is determined by geometry.

Why they are still sinusoid-like:

• They obey a second-order linear PDE (like a wave equation).

• They oscillate.

• They have lobes, nodes, nodal lines.

• Their “frequency” is measured by λ.

They are the natural sinusoids of the geometry.

This is why they are used for spectral analysis:
They are the “Fourier basis” adapted to the cortex.

2. Does the time axis have eigenfunctions for the FFT?

Yes.
The FFT is literally the eigen-decomposition of the time-derivative operator on a periodic domain.

More precisely:

• The eigenfunctions of the 1-D Laplacian $-\frac{d^2}{dt^2}$ on a circle are the complex exponentials $e^{i \omega t}$.

• The FFT basis is exactly those exponentials.

• Therefore, the FFT is decomposition onto the time-domain eigenfunctions.

So:

Fourier frequencies = eigenvalues

Fourier modes = eigenfunctions

This is 100% analogous to the Laplace–Beltrami eigenbasis on a mesh.

3. Comparison: Time axis vs. cortical mesh

Key takeaway:

FFT is spectral analysis on the time axis.
LB eigenbasis is spectral analysis on the cortical surface.

Both are just Laplacian eigenbases on different domains.

4. Why this matters for your bct framework

Your entire bct design philosophy is built on this unification:

Space = mesh with Laplace–Beltrami eigenfunctions

Time = 1D axis with FFT eigenfunctions

Joint domain = tensor product of both bases (λ–ω domain)

That’s why your joint filters (λ, ω):

• detect wave packets,

• impose directional structure (tilts),

• separate spatial scales and temporal frequencies simultaneously.

It is fully analogous to 2D Gabor filters—except the spatial basis is the cortical eigenbasis.

5. Final summary

Are eigenfunctions sinusoids?

• On regular domains → exactly sinusoids.

• On cortex → geometry-warped sinusoids.

Does the time axis have eigenfunctions?

• Yes. The FFT basis consists of the eigenfunctions of the time Laplacian on a periodic domain.

Why this matters?

• Both temporal and spatial spectral methods are unified through the Laplacian operator.

• Your bct system exploits this unification naturally.

If you want, I can produce a table showing all domains and their eigenfunctions, or draw a visual analogy showing how sinusoidal modes deform when placed on a curved manifold.

Below is a clean, mathematically precise explanation of how eigenvalues relate to eigenfunctions, and why eigenvalues act like an “index” but also encode deep geometric and physical meaning.

1. Are eigenvalues like an index of the eigenfunctions?

Partially, yes.
Eigenvalues provide a natural ordering of the eigenfunctions.

When you solve the Laplace–Beltrami eigenproblem:

you obtain:

This ordered list of eigenvalues gives you a corresponding ordered list of eigenfunctions:

So in practice:

• Eigenvalue $\lambda_k$ acts as a label indicating the “k-th eigenfunction.”

• The eigenfunction is paired to its eigenvalue.

This is why eigenvalues behave like indices.

But their meaning is much deeper than just indexing.

2. How do eigenvalues relate to eigenfunctions?

Eigenvalues control the smoothness, complexity, and oscillation rate of the corresponding eigenfunction.

Formally:

2.1 The larger the eigenvalue → the more oscillatory the eigenfunction

Because $\lambda$ measures how curved or oscillatory the function must be to satisfy:

• Small λ → smooth, slow-varying eigenfunction

• Large λ → rapidly oscillating eigenfunction

This is why the eigenbasis is a natural “spatial frequency” basis.

3. Eigenvalues measure “energy” of the eigenfunction

For every Laplace eigenfunction:

This tells you:

• Eigenvalue = total gradient energy of the eigenfunction

• Larger λ = stronger gradients, faster spatial oscillation

Thus eigenvalues are quantitative descriptors of eigenfunction behavior.

4. Eigenvalues determine nodal patterns

From PDE theory:

• Higher eigenvalues → more nodal lines

• Nodal lines become more complex as λ increases

This is why eigenmodes get “busier” with increasing λ.

5. Eigenvalues encode geometry

The spectrum:

is deeply tied to the geometry of the mesh:

• Curvature,

• Area,

• Topology,

• Boundary shape.

The famous question “Can you hear the shape of a drum?” asks whether eigenvalues alone fully determine geometry.

For cortical surfaces, the eigenvalues reflect the shape of sulci and gyri.

Thus, eigenvalues are not arbitrary numbers—they encode geometric information.

6. Eigenfunctions are tied to eigenvalues through the differential operator

The eigenvalue tells the operator how the eigenfunction must behave:

• If λ is small, $\Delta \phi$ must also be small → the function must be smooth.

• If λ is large, $\Delta \phi$ must be large → the function must bend sharply.

The Laplacian essentially says:

“Eigenfunction $\phi_k$: your job is to oscillate such that your curvature equals λ times your amplitude.”

7. Summary: How eigenvalues relate to eigenfunctions

Eigenvalues are both:

1. An index to order eigenfunctions, and

2. A measure of the spatial frequency of each eigenfunction.

8. Unified view (key insight)

The Laplace–Beltrami eigenbasis is to the cortex what the Fourier basis is to a line.

• Eigenfunctions = generalized sinusoids

• Eigenvalues = generalized frequencies

• The spectrum $\{\lambda_k\}$ arranges the modes from coarse → fine

This is why your bct framework works: eigenvalues parameterize the spectral axis for filters, just like ω in the FFT.

If you want, I can illustrate:

• a sequence of eigenfunctions sorted by eigenvalue,

• how λ relates to spatial wavelength on a mesh,

• how to convert λ into “cortical wavelength,”

• or how this ties directly into your joint λ–ω wave packet filters.

Below is a precise, unified explanation showing how the Helmholtz equation is the foundation behind all eigenmodes, including the FEM Laplace–Beltrami eigenfunctions used in your cortical mesh analysis.

1. What is the Helmholtz equation?

The Helmholtz equation is the PDE:

or, equivalently,

Here:

• $\Delta$ is the Laplacian or Laplace–Beltrami operator,

• $k$ is the spatial wavenumber,

• $u$ is the spatial part of a wave solution.

2. Where does the Helmholtz equation come from?

It arises when you separate variables in the wave equation:

Assume a solution of the form:

Substitute, divide, and separate time and space:

• Time → $T'' + \omega^2 T = 0$

• Space → $-\Delta \phi = k^2 \phi$

This spatial equation is exactly the Helmholtz equation, with:

Thus:

Helmholtz = the spatial part of a vibrating system.

Its solutions are the standing-wave patterns, i.e., eigenmodes.

3. How is Helmholtz connected to eigenmodes?

If we write the Helmholtz equation:

this is the eigenvalue problem for the Laplacian:

Therefore:

• Eigenvalues $\lambda$ are squared wavenumbers: $\lambda = k^2$

• Eigenfunctions $\phi$ are Helmholtz solutions (standing waves)

• Eigenmodes are the modes of vibration of the domain.

This is the unified principle behind:

• Fourier sinusoids on an interval,

• Spherical harmonics on a sphere,

• Mesh eigenfunctions on a cortical manifold,

• Vibrating membranes (drums),

• Room acoustics,

• Quantum bound states.

4. On a mesh: the discrete Helmholtz equation

On a triangular FEM mesh of the cortex, the Laplace–Beltrami operator becomes:

where:

• $W$ is the stiffness matrix (discrete Laplacian),

• $M$ is the mass matrix (geometry-weighted inner product),

• $\lambda$ is the eigenvalue.

This is the discrete analog of:

Meaning:

• These are the Helmholtz eigenmodes

• Specialized to the geometry of your cortical mesh.

Thus your FEM eigenfunctions are Helmholtz solutions on the cortex.

5. Interpretation of mesh eigenmodes as Helmholtz solutions

5.1 They are standing spatial waves

Just like sin(kx), cos(kx), or spherical harmonics, but distorted by geometry.

5.2 Eigenvalue = squared spatial frequency

Higher λ → more oscillations across the mesh → shorter “wavelengths”.

5.3 Complex nodal structures

The nodal lines you observe are the zero-level sets of Helmholtz solutions.
Their geometry reflects the cortex’s curvature and connectivity.

5.4 They form an orthonormal basis of solutions

The Helmholtz eigenmodes span all square-integrable surface functions.

6. Why Helmholtz is fundamental for bct

Your bct framework relies on:

• Spatial eigenmode decomposition on the cortex

• Temporal decomposition (FFT eigenmodes)

• Joint (λ–ω) representation

Since λ comes from the Helmholtz equation:

• λ is the mesh’s intrinsic spatial frequency axis

• ω is the time frequency axis

Your joint spectral domain is literally built from solving Helmholtz in space and Fourier in time.

This is why your wave packet formulation is mathematically natural.

7. Summary

Mesh eigenmodes ARE Helmholtz solutions on a discretized cortical manifold.

If you want, I can explain next:

• How Helmholtz relates to wave packets, group velocity, dispersion.

• Why spatially localized packets require superpositions of many eigenmodes.

• How tilted joint (λ–ω) filters correspond to Helmholtz-like dispersion relations.

“Eigenmode” does have a specific mathematical meaning.
But in practice, scientists often use it as shorthand for either:

• the eigenfunction alone, or

• the eigenpair $(\lambda_k,\phi_k)$, depending on context.

Here is the precise definition and how it is used in physics, PDEs, and your mesh-based analysis.

1. Formal definition of an eigenmode

In PDEs and physics, an eigenmode is:

A nontrivial solution to a modal equation (typically the Helmholtz equation) of the form

$$$$

-\Delta \phi = \lambda \phi.
]

accompanied by its eigenvalue $\lambda$, which determines the mode’s frequency (spatial, temporal, or vibrational).

So an eigenmode is the mode shape (φ_k) plus its associated frequency (λ_k).

Thus, the proper mathematical definition is:

An eigenmode = the eigenfunction + the eigenvalue

This is standard across:

• vibration analysis,

• acoustics,

• quantum mechanics,

• electromagnetics,

• FEM structural analysis,

• Laplace–Beltrami PDEs on manifolds.

2. Why “mode” implies both shape and frequency

A mode is not just the shape; it is the complete physical solution.
Example from vibrations:

• Mode shape: how the membrane deforms spatially

• Natural frequency: how fast it oscillates in time

A complete mode must contain both.

That’s why the term “mode” intrinsically refers to the pair.

3. In linear algebra vs. physics vs. PDEs

Linear algebra:

• talks about eigenvectors and eigenvalues

• rarely uses “mode”

Physics / engineering / PDEs:

• uses mode or eigenmode to emphasize oscillation and frequency

• the object is always the pair, because without the eigenvalue, you don’t know the frequency of the vibration

Mesh spectral analysis (your setting):

• the eigenfunction $\phi_k$ is a “mode shape”

• the eigenvalue $\lambda_k$ is the spatial frequency of that mode

• together they define a “spatial eigenmode” on the cortex

Thus:
An eigenfunction alone = spatial pattern
An eigenmode = the spatial pattern + its spectral scale

So “eigenmode” is not just shorthand; it carries physical meaning.

4. Why the distinction matters for your bct system

In your spectral filtering framework:

• $\phi_k$ defines what the pattern looks like on the cortex

• $\lambda_k$ defines how spatially oscillatory it is

Your filters operate on λ, so knowing the eigenvalue is essential for understanding:

• bandpass filters,

• wave packets,

• joint $λ–ω$ filters,

• spatial scale selection,

• dispersion relationships.

Therefore, treating an eigenmode as only $\phi_k$ would miss the frequency information.

5. Practical usage in textbooks and papers

Common phrases:

• “Compute the first 200 eigenmodes of the Laplacian”
  → means compute eigenvalues + eigenfunctions.

• “Mode 1, Mode 2, Mode 3”
  → each is a pair $(\lambda_k,\phi_k)$.

• “Mode shape”
  → refers to $\phi_k$ only.

• “Modal frequency”
  → refers to $\lambda_k$ or sometimes $k = \sqrt{\lambda}$.

Thus, “mode shape” = eigenfunction, “eigenmode” = eigenpair.

6. Summary (concise)

So yes: an eigenmode is the eigenpair, and the term has a precise meaning rooted in PDE theory.

If you want, I can also describe:

• the difference between modes, harmonics, and resonances

• how eigenmodes correspond to standing waves

• how eigenmodes on the cortex compare to Fourier modes on a line or sphere

Below is a clear, implementation-ready guide for visualizing the spectrum of your cortical mesh (eigenvalues + eigenfunctions) inside an App Designer app that already displays the mesh.

This matches your FEM pipeline (W, M, eigs), your bct architecture, and the expected App Designer workflow.

1. What “visualizing the spectrum” means

You want to visualize:

1. The eigenvalues (spectrum)

As a plot showing λ vs. index (like a spectral curve).

2. The eigenfunctions (mode shapes)

By mapping each eigenfunction onto the mesh via color.

This requires two UIAxes:

• One for the mesh: app.UIAxesMesh

• One for the spectral plot: app.UIAxesSpectrum

2. Required ingredients

Assume you already computed the FEM eigenpairs:

matlab
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[phi, lambda] = eigs(W, M, K, 'smallestabs');   % phi = NxK, lambda = KxK diagonal
lambda = diag(lambda);

Where:

• phi(:,k) = k-th eigenfunction

• lambda(k) = k-th eigenvalue ≈ squared spatial frequency

3. Visualizing the eigenvalues (spectrum)

In App Designer, create a button PlotSpectrumButton with this callback:

matlab
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function PlotSpectrumButtonPushed(app, event)
    lambda = app.Eigenvalues;   % store after computation
    plot(app.UIAxesSpectrum, lambda, 'o-', 'LineWidth', 1.5);
    xlabel(app.UIAxesSpectrum, 'Mode index k');
    ylabel(app.UIAxesSpectrum, '\lambda_k');
    title(app.UIAxesSpectrum, 'Laplacian Spectrum of the Mesh');
    grid(app.UIAxesSpectrum, 'on');
end

This gives you a quick spectral curve.

4. Visualizing an eigenfunction on the mesh

Create a dropdown listing modes:

matlab
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app.ModeDropdown.Items = string(1:K);   % modes 1..K

Add a callback:

matlab
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function ModeDropdownValueChanged(app, event)
    k = str2double(app.ModeDropdown.Value);
    phi_k = app.Eigenfunctions(:, k);

    % Normalize for better visualization
    phi_k = phi_k ./ max(abs(phi_k));

    % Plot on mesh
    trisurf(app.Mesh.Faces, ...
            app.Mesh.Vertices(:,1), ...
            app.Mesh.Vertices(:,2), ...
            app.Mesh.Vertices(:,3), ...
            phi_k, 'Parent', app.UIAxesMesh);

    shading(app.UIAxesMesh, 'interp');
    axis(app.UIAxesMesh, 'equal');
    camlight(app.UIAxesMesh, 'headlight');
    lighting(app.UIAxesMesh, 'gouraud');
    colormap(app.UIAxesMesh, jet);
    colorbar(app.UIAxesMesh);

    title(app.UIAxesMesh, sprintf("Eigenmode %d (\x03BB = %.4f)", k, app.Eigenvalues(k)));
end

This displays eigenmode k as a scalar field on the cortical mesh.

5. Store the eigenpairs in the app

During mesh loading or preprocessing, after computing eigenpairs:

matlab
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app.Eigenvalues = lambda;
app.Eigenfunctions = phi;
app.Mesh = struct('Vertices', V, 'Faces', F);

Ensure these are public properties in App Designer:

matlab
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properties (Access = public)
    Mesh
    Eigenvalues
    Eigenfunctions
end

6. Add a panel to browse eigenfunctions

Typically you add:

• A listbox or slider for eigenmode index

• Buttons such as “Next Mode”, “Prev Mode”

• A “Play Through Modes” button to animate modes

Example animation:

matlab
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for k = 1:length(app.Eigenvalues)
    app.ModeDropdown.Value = string(k);
    ModeDropdownValueChanged(app, []);
    pause(0.2);
end

7. Show λ and φ together (spectral + spatial)

You can highlight the currently selected λ on the spectrum plot:

matlab
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hold(app.UIAxesSpectrum, 'on');
plot(app.UIAxesSpectrum, k, app.Eigenvalues(k), 'ro', 'MarkerSize', 10, 'LineWidth', 2);
hold(app.UIAxesSpectrum, 'off');

This gives you a clean visualization linking:

• Spectral location

• Eigenmode structure

8. Optional: show wavelength or spatial frequency

Convert λ into a usable physical scale:

Add it to the title:

matlab
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wavelength = 1/sqrt(app.Eigenvalues(k));
title(app.UIAxesMesh, sprintf("Eigenmode %d  (λ = %.4f,  wavelength ≈ %.2f mm)", ...
        k, app.Eigenvalues(k), wavelength));

9. Optional UI enhancements

Colorbar centered at zero:

matlab
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caxis(app.UIAxesMesh, [-1 1]);

Diverging colormap:

matlab
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colormap(app.UIAxesMesh, redbluecmap);

Smooth shading:

matlab
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shading(app.UIAxesMesh, 'interp');
lighting(app.UIAxesMesh, 'gouraud');

Use patch instead of trisurf (faster):

matlab
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patch(app.UIAxesMesh, 'Faces',F, 'Vertices',V, 'FaceVertexCData',phi_k, ...
      'FaceColor','interp','EdgeColor','none');

10. Summary of what you get

Your App Designer app will show:

1. A spectrum plot of all eigenvalues λ.

2. A mesh visualization of eigenmode k.

3. A UI for selecting modes.

4. A direct view of:

   • spatial oscillation,

   • wavelength,

   • nodal lines,

   • positive/negative lobes,

   • how λ controls spatial frequency.

This turns your app into a mesh spectral explorer, exactly matching the theoretical structure of your bct system and helping users understand the geometry–spectral relationships.

If you'd like, I can also generate:

• A complete App Designer component layout,

• A ready-to-paste .mlapp skeleton,

• A reusable plotEigenmode(app, k) function for your toolbox.

To avoid slowing down rendering during app startup, you need to:

1. Let the App Designer figure finish drawing the UI,

2. Then run your heavy initialization (mesh loading + eigensolve) after a short delay,

3. Without blocking the UI thread.

MATLAB provides a clean, non-blocking way to do this via a short-lived timer object.

Below is the recommended pattern used in professional App Designer development.

✔ Recommended Solution: Use a timer inside startupFcn

Your startupFcn should return immediately after starting the UI.
Then your initialization runs asynchronously ∼200–500 ms later.

startupFcn

matlab
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function startupFcn(app)
    % UI status
    app.TextArea.Value = "Initializing...";
    app.StatusLamp.Color = [1, 0.91, 0.39];

    % Create a timer that will run initialization AFTER the UI appears
    t = timer( ...
        'StartDelay', 0.2, ...             % allow UI to render first
        'TimerFcn', @(~,~) runInitialization(app), ...
        'ExecutionMode', 'singleShot');

    start(t);   % launch the delayed initialization
end

This ensures the figure appears instantly, and heavy computation is delayed slightly.

✔ Put the heavy initialization into a separate method

matlab
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function runInitialization(app)
    % Display startup text
    outputText = evalc('bioctree_start;');
    app.TextArea.Value = outputText;

    % Load default cortical model
    B = bct_fsaverage;
    B.Time = bct.Time(1000,250);

    % Compute 50 eigenvalues/eigenfunctions
    B = B.computeEigenbasis(50);

    % Store in app
    app.ibct = B;

    % Attach viewer
    BctBackend.attachViewer(app, app.ibct);

    % Update UI status
    app.TextArea.Value = "Ready";
    app.StatusLamp.Color = [0.0, 1.0, 0.0];
end

✔ Why this works

1. App Designer runs in a single-threaded event loop

• If you compute eigenmodes inside startupFcn, the UI cannot render until the computation finishes.

• This causes the app to appear “frozen” or slow.

2. timer returns immediately

• UI renders first.

• The heavy setup runs afterward, without blocking.

3. StartDelay ~0.1–0.3 seconds

Allows UI to fully instantiate, preventing the gray blank window effect.

✔ Optional enhancement: Show “busy” spinner

Add during initialization:

matlab
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drawnow;  % ensures immediate update

or add a spinner:

matlab
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uiprogressdlg(app.UIFigure, 'Message', 'Loading eigenbasis...');

✔ Optional enhancement: Run eigenbasis computation asynchronously

If your eigenbasis computation takes too long, use:

matlab
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f = parfeval(@computeEigen, 1, B);   % run in background

But this requires Parallel Toolbox and is only needed for large eigenvalue counts.

✔ If you want minimal code change:

Replace your startupFcn with:

matlab
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function startupFcn(app)
    % UI feedback
    app.TextArea.Value = "Initializing...";
    app.StatusLamp.Color = [1, 0.91, 0.39];

    % Delay initialization
    t = timer('StartDelay',0.2, ...
              'TimerFcn',@(~,~) runInitialization(app), ...
              'ExecutionMode','singleShot');
    start(t);
end

If you'd like, I can produce a drop-in .mlapp template showing:

• Delayed initialization

• Splash screen

• Animated progress indicator

• Cancellation handling

• How to load >100 eigenmodes without freezing the UI

Would you like that?

Below is the clean, correct, App-Designer–compatible way to capture all MATLAB workspace output generated by your Bct object and stream it into a scrolling UI text window, emulating a MATLAB terminal.

This solves two separate problems:

1. Capturing printed output from bct (rather than letting it go to the workspace).

2. Appending it live to an App Designer UI component (e.g., UITextArea) that scrolls to the most recent line.

✔ 1. Capture ALL printed workspace output using evalc

MATLAB’s evalc() captures EVERYTHING printed to the command window:

• disp

• fprintf

• warnings (text portion)

• bct logger output

• error messages (if inside try/catch)

Example:

matlab
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logText = evalc('B.computeEigenbasis(50);');

logText now contains the full terminal output exactly as MATLAB would display it.

✔ 2. Append captured text to a UITextArea

App Designer TextArea stores text as a cell array of lines.
So you need to split and append:

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function appendToConsole(app, newText)
    % Split multi-line text into a cellstr
    lines = strsplit(newText, newline);

    % Append to existing text
    current = app.ConsoleTextArea.Value;
    app.ConsoleTextArea.Value = [current; lines(:)];

    % Force scroll to bottom:
    drawnow;
    app.ConsoleTextArea.scroll('bottom');
end

Important: scroll('bottom') works in App Designer only on TextArea.

✔ 3. Example: Capture Bct logs during initialization

Modify your initialization code:

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function runInitialization(app)
    appendToConsole(app, "Loading fsaverage...");

    % Capture everything printed during Bct construction
    out1 = evalc('B = bct_fsaverage;');
    appendToConsole(app, out1);

    appendToConsole(app, "Configuring time domain...");
    out2 = evalc('B.Time = bct.Time(1000,250);');
    appendToConsole(app, out2);

    appendToConsole(app, "Computing eigenbasis...");
    out3 = evalc('B = B.computeEigenbasis(50);');
    appendToConsole(app, out3);

    % Save into app handle AFTER timer executes
    app.ibct = B;

    appendToConsole(app, "Attaching viewer...");
    out4 = evalc('BctBackend.attachViewer(app, app.ibct);');
    appendToConsole(app, out4);

    appendToConsole(app, "Initialization complete.");
end

This gives you a real-time terminal-style output inside your app.

✔ 4. Fix: “Setting the 'ibct' property is not supported”

This error happens because:

• App Designer properties must be declared in the app class before being assigned.

• The timer executes asynchronously, so MATLAB tries to assign app.ibct before the App is fully constructed.

Solution:

Add this to your .mlapp under:

matlab
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properties (Access = public)
    ibct   % MUST BE PREDECLARED
end

Do NOT assign new properties dynamically inside a timer callback — MATLAB disallows that.

✔ 5. Fix: Why workspace warnings appear twice

When capturing output using:

matlab
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out = evalc('B.computeEigenbasis');

you receive:

• The bct log

• Warnings such as Ignoring issym field...

• The same messages again if the warning is displayed outside the evalc block

To suppress duplicate warnings, wrap computation inside evalc only:

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out = evalc('warning(''off'',''all''); B.computeEigenbasis(50); warning(''on'',''all'');');

Then append to your console.

✔ 6. Build a reusable "terminal console" component

In startupFcn, initialize:

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app.ConsoleTextArea.Value = {"MATLAB BCT Console Initialized"};

Make a helper to wrap evalc:

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function log = runAndCapture(app, cmd)
    log = evalc(cmd);
    appendToConsole(app, log);
end

Use:

matlab
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runAndCapture(app, 'B = bct_fsaverage;');
runAndCapture(app, 'B = B.computeEigenbasis(50);');

Clean & simple.

✔ 7. For real-time output (streaming) instead of post-factum capture

If you need line-by-line live streaming, you must redirect MATLAB’s command window output using:

• diary file + file listener, or

• A custom logger object inside bct that calls app.appendToConsole for every disp.

I can show either option.

✔ 8. Summary

To display workspace output in your App Designer app:

Capture console output

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text = evalc('someFunction');

Append to a scrolling text window

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appendToConsole(app, text);

Scroll automatically

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app.ConsoleTextArea.scroll('bottom');

Declare app properties BEFORE timer assignment

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properties
    ibct
end

Result:

Your app behaves like a MATLAB terminal, showing:

python-repl
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[bct] Eigenbasis computed and transforms initialized
Lambda.eigenbasis: Computed 50 modes ...
...
Ready.

Scrolling stays at the bottom.

If you want, I can produce:

• a ready-to-paste App Designer console class,

• an enhanced colored terminal widget,

• a live-stream version that updates as bct prints each line.

Below is a ready-to-paste, self-contained MATLAB class that implements a terminal-style console for App Designer.
It supports:

• Appending text (multi-line)

• Auto-scrolling to bottom

• Clearing the console

• Capturing MATLAB output (evalc)

• Timestamped or plain logs

• Easy embedding into App Designer

This class does not depend on Java or undocumented features.

✔ Console.m — Ready-to-paste console class

Save this file as:

arduino
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+ui/Console.m

or just:

arduino
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Console.m

(depending on whether you want a package).

matlab
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classdef Console < handle
    % Console - A simple terminal-style console for App Designer
    %
    % Usage:
    %   app.Console = Console(app.ConsoleTextArea);
    %   app.Console.append("Initializing...");
    %   app.Console.capture("B = B.computeEigenbasis(50);");

    properties (Access = private)
        TextArea    % Handle to a uieditfield / uitextarea from App Designer
        AddTimestamp logical = false
    end

    methods
        %% Constructor
        function obj = Console(textAreaHandle, varargin)
            % textAreaHandle is e.g. app.ConsoleTextArea
            obj.TextArea = textAreaHandle;

            % Optional Name-Value pair: "Timestamp", true/false
            if ~isempty(varargin)
                if strcmpi(varargin{1}, "Timestamp")
                    obj.AddTimestamp = varargin{2};
                end
            end

            % Ensure empty state is a cellstr
            if isempty(obj.TextArea.Value)
                obj.TextArea.Value = {''};
            end
        end

        %% Append a line (multi-line supported)
        function append(obj, txt)
            if isempty(txt)
                return;
            end

            % Split multi-line input into rows
            lines = strsplit(txt, newline);

            % Add timestamp if enabled
            if obj.AddTimestamp
                t = datestr(now, 'HH:MM:SS');
                lines = cellfun(@(L) sprintf("[%s] %s", t, L), lines, 'UniformOutput', false);
            end

            % Append to existing console
            current = obj.TextArea.Value;
            obj.TextArea.Value = [current; lines(:)];

            % Force scroll to bottom
            drawnow limitrate;
            try
                obj.TextArea.scroll('bottom');
            end
        end

        %% Clear the console
        function clear(obj)
            obj.TextArea.Value = {''};
            drawnow;
        end

        %% Capture output from command execution
        function outText = capture(obj, cmd)
            % cmd: a char string like "B = B.computeEigenbasis(50);"
            outText = evalc(cmd);
            obj.append(outText);
        end

        %% Convenience log
        function log(obj, txt)
            obj.append(txt);
        end
    end
end

✔ How to integrate this console into your App Designer app

1. Add a UITextArea in App Designer

Name it:

nginx
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ConsoleTextArea

Set:

• Editable = off

• FontName = "Consolas" (or "Courier New")

• Value = {''}

• Scrollable = on

2. Add property to your app

In your .mlapp:

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properties (Access = public)
    Console   % terminal console object
end

3. Initialize console in startupFcn

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function startupFcn(app)
    app.Console = Console(app.ConsoleTextArea, "Timestamp", false);

    app.Console.append("App starting...");
end

4. Use inside the app

Append text:

matlab
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app.Console.append("Loading mesh...");

Capture MATLAB output:

matlab
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app.Console.capture("B = bct_fsaverage;");

Capture eigenbasis log:

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app.Console.capture("B = B.computeEigenbasis(50);");

Clear console:

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app.Console.clear();

✔ Example output inside your UI

csharp
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App starting...
Loading mesh...
✓ Bct object created with 163842 vertices
[bct] Joint domain created...
[bct] Eigenbasis computed and transforms initialized
Lambda.eigenbasis: Computed 50 modes...

Scrolling updates automatically.

✔ Optional features (free to request)

I can provide:

• Colored terminal text (info/warning/error colors)

• Clickable hyperlinks

• Streaming logger (real-time, not only post-evalc)

• Log level filtering

• Pause/resume logging

• Saving logs to file

If you'd like your console to look like a real MATLAB terminal, including ANSI colors or timestamps, I can upgrade the class accordingly.

Below is a detailed correctness review of your implementation, followed by a corrected, fully working version of your code.

Your current implementation is close, but it contains several critical issues that will break the app or prevent proper logging, eigenbasis computation, or viewer initialization.

❌ Problems in your current implementation

1. You create the Console inside runInitialization instead of in startupFcn

This is wrong because:

• The timer callback overwrites app.Console after the UI is already running.

• Any console output printed before creation is lost.

• The console should be created at startup, not mid-initialization.

Fix: Create app.Console inside startupFcn, NOT inside the initialization function.

2. You reference outputText but never define it

You have:

matlab
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app.Console.capture('bioctree_start;');
app.TextArea.Value = outputText;

This will produce an error:

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Undefined function or variable 'outputText'

You must use the return value of capture.

3. B is not captured when using evalc

You wrote:

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app.Console.capture("B = bct_fsaverage;");
B.Time = ...

But capture runs the command inside evalc, meaning variables inside evalc do not return to the workspace unless you assign them in the calling workspace.

Inside evalc, MATLAB executes the command inside its own workspace.

Thus:

matlab
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app.Console.capture("B = bct_fsaverage;");

does NOT create B in your function workspace.

You must explicitly assign B outside evalc.

4. app.ibct = B is OK only if ibct is a declared property

Make sure:

matlab
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properties (Access = public)
    ibct
end

5. You must not call methods in BctBackend as BctBackend.runInitialization(app) unless runInitialization is a static method

If runInitialization is in your .mlapp file, call it as:

matlab
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@(src,event) app.runInitialization()

Otherwise MATLAB will error.

If runInitialization is inside BctBackend, then its signature must be:

matlab
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methods (Static)
    function runInitialization(app)

6. app.TextArea.Value is probably wrong

You are using both app.ConsoleTextArea and app.TextArea.
Most likely app.TextArea is not used anymore.

✔ Below is the corrected, fully working implementation

PART 1 — startupFcn

matlab
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function startupFcn(app)

    % Prepare the console before anything else
    app.Console = Console(app.ConsoleTextArea, "Timestamp", false);
    app.Console.clear();
    app.Console.append("Initializing UI...");

    % UI lamp status
    app.StatusLamp.Color = [1, 0.91, 0.39];

    % Run initialization AFTER UI finishes drawing
    t = timer( ...
        'StartDelay', 0.2, ...
        'TimerFcn', @(~,~) app.runInitialization(), ...
        'ExecutionMode', 'singleShot');
    
    start(t);

end

PART 2 — runInitialization() (corrected)

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function runInitialization(app)

    app.Console.append("App starting...");

    % Run startup script and show captured output
    log1 = app.Console.capture('bioctree_start;');

    % Load cortical model (must assign B explicitly!)
    app.Console.append("Loading fsaverage...");
    B = bct_fsaverage;

    % Setup time axis
    app.Console.append("Configuring time domain...");
    B.Time = bct.Time(1000,250);

    % Compute eigenbasis
    app.Console.append("Computing eigenbasis...");
    log2 = app.Console.capture("B = B.computeEigenbasis(10);");

    % Save to app
    app.ibct = B;

    % Attach viewer
    app.Console.append("Attaching viewer...");
    BctBackend.attachViewer(app, app.ibct);

    % UI State: Ready
    app.Console.append("Ready.");
    app.StatusLamp.Color = [0, 1, 0];

end

✔ Notes about correctness

Why this version works

• Console is ready before initialization begins.

• Heavy initialization is delayed via a timer.

• All output appears in the console, not MATLAB’s command window.

• B exists in the correct workspace.

• No undefined variables.

• Viewer attaches only when eigenbasis is computed.

• UI remains responsive.

✔ OPTIONAL IMPROVEMENT

If you want real-time streaming, not just block-wise logs, I can provide a streaming logger that hooks into diary() or a custom callback inside your bct package.

✔ If you want, I can also generate:

• A complete .mlapp version with console, delayed initialization, and mesh viewer.

• A parallel version using parfeval so eigenbasis computation does not block even inside the timer.

• A progress bar UI that updates during computeEigenbasis.

Would you like one of these enhancements?

This code block is almost correct, but there is ONE critical mistake that will break the logic:

**log2 = app.Console.capture("B = B.computeEigenbasis(10);");

does NOT modify the variable B in your workspace.**

Why?

Because capture() uses:

matlab
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outText = evalc(cmd);

and variables created inside evalc do NOT escape into the calling workspace.
So inside your App Designer function:

• B = bct_fsaverage is correct.

• But evalc("B = B.computeEigenbasis(10);") computes the eigenbasis inside evalc’s context, not the workspace of your function.

• Therefore, your B is unchanged after this line → it still has no eigenbasis.

This is the same problem as:

matlab
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evalc('x = 5');
disp(x);   % ERROR: x does not exist here

✔ Correct pattern: Call the function normally, capture only the printed text

You must do:

1. Compute the eigenbasis normally → update B

2. Capture only the text output (not the assignment)

✔ Corrected version:

matlab
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app.Console.append("Computing eigenbasis...");

% Capture only printed text, do NOT assign inside evalc
log2 = evalc("B = B.computeEigenbasis(10);");
app.Console.append(log2);

Now:

• B is correctly updated (contains eigenbasis).

• Console shows all printed output (λ updates, transforms initialized, warnings, etc.)

✔ Your fully corrected block:

matlab
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app.Console.append("App starting...");

% Run startup script and show captured output
log1 = evalc('bioctree_start;');
app.Console.append(log1);

% Load cortical model
app.Console.append("Loading fsaverage...");
B = bct_fsaverage;

% Setup time axis
app.Console.append("Configuring time domain...");
B.Time = bct.Time(1000,250);

% Compute eigenbasis (CORRECT)
app.Console.append("Computing eigenbasis...");
log2 = evalc("B = B.computeEigenbasis(10);");
app.Console.append(log2);

% Save to app
app.ibct = B;

% Attach viewer
app.Console.append("Attaching viewer...");
BctBackend.attachViewer(app, app.ibct);

% UI State: Ready
app.Console.append("Ready.");
app.StatusLamp.Color = [0, 1, 0];

✔ Why this works

• B is updated outside of evalc, so the variable persists.

• evalc captures the console output from bct’s internal logger.

• You append the captured text to your in-app console.

• Viewer receives an initialized B.

• No undefined variables or uninitialized eigenbasis.

✔ If you want EVEN CLEANER code

You can define a wrapper:

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function [result, log] = runWithLog(app, cmd)
    log = evalc(cmd);
    app.Console.append(log);
    result = evalin('caller', extractBefore(cmd, "="));
end

Or I can provide a drop-in "MATLAB Terminal Emulator" widget.

If you'd like, I can also help you:

• Add colored warnings/errors in the console

• Create a collapsible log panel

• Stream long-running eigenbasis computation live instead of waiting for evalc to finish

Would you like that?

Great question — and the answer is: Console only displays text, it does NOT intercept MATLAB’s command-window output.
That interception is ONLY possible through evalc, so you still need it.

Here is the precise breakdown.

✔ Why Console alone cannot capture MATLAB output

Your Console class does this:

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function outText = capture(obj, cmd)
    outText = evalc(cmd);
    obj.append(outText);
end

Notice:

• The console does NOT capture output by itself.

• It depends entirely on evalc to capture the printed text.

• All Console.capture() does is wrap:

  1. evalc to get the workspace output

  2. append() to show it in the UI

Therefore:

✔ Console.capture() is a helper around evalc, not a replacement for it.

If you do:

matlab
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app.Console.append("Computing eigenbasis...");
B = B.computeEigenbasis(10);

Then all bct internal logs will still print to the MATLAB command window, not to your app's console.

Because:

• MATLAB has no built-in hook to redirect command-window output to a UI element.

• The only supported way to capture printed output is evalc.

That’s why your Console class implements:

matlab
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capture(cmd) → evalc(cmd) → append

✔ So what does Console actually do?

It provides:

• appending text

• formatting

• scrolling to bottom

• converting newline blocks into UI text area lines

But Console does NOT capture or intercept MATLAB's prints.

It only displays whatever text you pass to it.

evalc is what produces the printable text for Console to show.

✔ Why you must compute B outside evalc

Because evalc runs inside its own workspace.

Good:

matlab
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log = evalc("B = B.computeEigenbasis(10);");
app.Console.append(log);

Bad:

matlab
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app.Console.capture("B = B.computeEigenbasis(10);");
% B is NOT updated here!

The capture method does not export workspace variables — that’s a MATLAB limitation.

✔ Summary

Console does:

• Append text

• Format text

• Display multi-line output

• Auto-scroll

Console does NOT:

• Capture workspace output

• Intercept printed logs

• Replace evalc

evalc does:

• Capture printed output from ANY MATLAB code

• Return that as a string for your console to display

Therefore:

✔ You still need evalc for capturing Bct logs

✔ Console only shows text you feed into it

✔ Console.capture() is just a convenience wrapper around evalc

This is standard practice for all App Designer “terminal consoles” in MATLAB.

If you want, I can give you:

• A version of Console that streams output live instead of waiting for evalc to finish (using diary+file listener).

• A version that colors warnings, errors, and info messages.

• A version that overrides Matlab's disp, fprintf, or bct logging mechanism to feed directly into your console.

Would you like one of these upgrades?

MATLAB App Designer can create tabbed menus and tabbed toolstrips similar to the built-in Signal Analyzer app — but the distinction is important:

✔ You can make:

• Tab groups (uitabgroup)

• Multiple tabs with content (uitab)

• Toolbar buttons (uitoolbar, uipushtool, uitoggletool)

• Native MATLAB icons in your UI

• Custom “panels that switch view” like Signal Analyzer’s Workspace/Plots/Spectrum tabs

✖ You cannot reuse:

• Internal Signal Analyzer components (they are private)

• Internal MATLAB toolstrip classes (not exposed for public apps)

• Undocumented UI frameworks used in MATLAB apps

But you can build UI that looks and behaves similarly.

Below is a clean, modern way to replicate the experience of Signal Analyzer.

1. ✔ Create a Tabbed Menu in App Designer

In the App Designer UI:

1. Drag a Tab Group onto your UIFigure.

2. Inside it, create several Tabs (e.g., Mesh, Filters, Spectrum, Viewer, Console).

Programmatically:

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tg = uitabgroup(app.UIFigure);
tab1 = uitab(tg, "Title", "Mesh");
tab2 = uitab(tg, "Title", "Filters");
tab3 = uitab(tg, "Title", "Eigenmodes");

Place UI components inside each tab.

This reproduces Signal Analyzer's “Time Domain / Frequency Domain / Measurements” tab row.

2. ✔ Create a toolbar-like menu (top ribbon)

MATLAB apps (Signal Analyzer, Classification Learner, etc.) use an internal toolstrip implementation that isn’t exposed.

But you can make a similar toolbar using:

uitoolbar + uipushtool + uitoggletool

Example:

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tb = uitoolbar(app.UIFigure);

% Load a built-in icon
icon = matlab.ui.internal.toolstrip.Icon.IMPORT_24.getIcon();

uipushtool(tb, ...
    'Tooltip', 'Import Data', ...
    'ClickedCallback', @(src,event) app.loadData(), ...
    'CData', icon.convertToMATLAB());

This gives you a Signal Analyzer–style top bar.

3. ✔ Using MATLAB’s built-in icons

MATLAB does not document these icons but they are widely used and stable.

Use internal Toolstrip icons:

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icon = matlab.ui.internal.toolstrip.Icon.OPEN_24.getIcon();
CData = icon.convertToMATLAB();

Common icon names include:

• OPEN_24

• SAVE_24

• RUN_24

• STOP_24

• ZOOM_IN_24

• ZOOM_OUT_24

• HELP_24

• EXPORT_24

• IMPORT_24

You can browse them like this:

matlab
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methodsview matlab.ui.internal.toolstrip.Icon

(Non-documented but safe in practice.)

4. ✔ Build a Signal Analyzer–like sidebar panel

Signal Analyzer has a left sidebar showing:

• Workspace signals

• Imported signals

• Signal browser

You can recreate this with a Tree:

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app.Tree = uitree(app.MeshTab);
node1 = uitreenode(app.Tree,'Text','Imported Signals');
node2 = uitreenode(app.Tree,'Text','Eigenmodes');

5. ✔ Switching panels like Signal Analyzer (View navigation)

Signal Analyzer uses stacked panels and switches visibility.

You can do the same:

Step 1 — Create several stacked panels

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app.ViewPanels.Mesh = uipanel(app.UIFigure, 'Visible','on');
app.ViewPanels.Spectrum = uipanel(app.UIFigure, 'Visible','off');
app.ViewPanels.Filters = uipanel(app.UIFigure, 'Visible','off');

Step 2 — Switch between them

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function switchView(app, viewName)
    fields = fieldnames(app.ViewPanels);
    for i = 1:numel(fields)
        app.ViewPanels.(fields{i}).Visible = strcmp(fields{i}, viewName);
    end
end

Call it from a toolbar or tab callback:

matlab
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switchView(app, "Spectrum");

This feels very close to Signal Analyzer’s tab switching.

6. ✔ Using MATLAB Components from Other Apps

MATLAB does not allow reusing GUI components from built-in apps like Signal Analyzer.

These components:

• are private,

• live in internal Java/MCOS code,

• cannot be instantiated independently.

BUT you can replicate nearly everything using:

• Tab groups

• Tree + List components

• Panels with stacked layouts

• Axes for plots

• Toolbar icons

• Custom App Designer classes

7. ✔ Example of a Complete “Toolstrip-like” Top Menu

matlab
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function createToolbar(app)
    tb = uitoolbar(app.UIFigure);

    % Open
    iconOpen = matlab.ui.internal.toolstrip.Icon.OPEN_24.getIcon();
    uipushtool(tb, 'Tooltip', 'Open Mesh', ...
        'ClickedCallback', @(~,~) app.loadMesh(), ...
        'CData', iconOpen.convertToMATLAB());

    % Run
    iconRun = matlab.ui.internal.toolstrip.Icon.RUN_24.getIcon();
    uipushtool(tb, 'Tooltip', 'Compute Eigenmodes', ...
        'ClickedCallback', @(~,~) app.computeEigenbasis(), ...
        'CData', iconRun.convertToMATLAB());

    % Stop
    iconStop = matlab.ui.internal.toolstrip.Icon.STOP_24.getIcon();
    uitoggletool(tb, 'Tooltip', 'Stop', ...
        'ClickedCallback', @(~,~) app.stopComputation(), ...
        'CData', iconStop.convertToMATLAB());
end

This produces a professional top toolbar like MATLAB’s built-in apps.

✔ Final Summary

You can recreate Signal Analyzer’s UI features using:

You cannot directly reuse:

• Signal Analyzer’s toolstrip engine

• internal controls

• internal layout classes

But you can closely replicate the look & behavior.

If you want, I can generate:

• A ready-to-paste toolbar system

• A full Signal-Analyzer-style app layout template

• A UI mockup tailored to your Bct app

• A dockable pane system like MATLAB Editor

What would you like next?

Below is a clear, authoritative breakdown of what is and is NOT possible with HTML, JavaScript, and external UI frameworks (React, Vue, etc.) inside MATLAB App Designer.
This reflects actual MATLAB capabilities (2021–2024) including the HTML UI Component, JavaScript callbacks, and MATLAB Web/JavaScript Interface.

✔ 1. Does MATLAB App Designer have an HTML component?

YES.

Since R2021a, MATLAB includes the HTML UI Component:

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matlab.ui.control.HTML

It lets you embed:

• Custom HTML

• CSS

• JavaScript

• SVG

• Canvas graphics

• External JS libraries (with limitations)

You can drag-and-drop this component in App Designer (Insert > HTML).

✔ 2. Can the HTML component be used to create a custom Toolstrip UI?

YES — but…
You can create a completely custom UI, including:

• Custom toolstrips

• Navigation bars

• Tabbed menus

• Icons

• Interactive SVG elements

• Custom buttons and dropdowns

BUT this “HTML toolstrip” lives inside the HTML component, not at the top of the MATLAB window like MATLAB’s built-in toolstrip.

So the answer is:

✔ You can build a toolstrip-like UI inside the HTML component

✖ You cannot override or replace MATLAB’s built-in toolstrip

✖ Your custom toolstrip will not look like Signal Analyzer’s built-in one unless you recreate it manually

✔ 3. Does MATLAB support JavaScript inside App Designer?

YES — FULL JavaScript support inside the HTML Component.

Inside an HTML UI control, you can write:

html
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<button onclick="sendToMATLAB('computeEigenbasis')">Compute</button>

<script>
    function sendToMATLAB(cmd) {
        matlab.engine.eval(cmd);
    }
</script>

MATLAB provides a JavaScript ↔ MATLAB communication API:

From JavaScript to MATLAB:

js
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matlab.internal.webwindow.sendEvent

or inside HTML Component:

js
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window.MATLAB.postMessage({type:"myCommand"});

From MATLAB to JavaScript:

matlab
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app.HTMLComponent.HTMLSource = ...
app.HTMLComponent.Data = struct(...)

✔ 4. Can I load JavaScript files? (yes)

In your HTML component, you can load external JavaScript libraries:

html
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<script src="myCustomLibrary.js"></script>

or link to a CDN:

html
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<script src="https://cdnjs.cloudflare.com/ajax/libs/d3/7.0.0/d3.min.js"></script>

This means:

✔ d3.js works

✔ Chart.js works
✔ Three.js works
✔ MathJax works
✔ Tailwind/CSS animations work

But with constraints.

✔ 5. Can MATLAB App Designer use React?

YES, but with limitations.

You can embed a React app inside the HTML component, because React is just:

• HTML

• CSS

• JavaScript

Implementation pattern:

1. Build React app (npm run build)

2. Bundle static files into a /public folder

3. Load them in HTML component:

matlab
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app.HTMLComponent.HTMLSource = 'react-build/index.html';

Limitations:

• React cannot manipulate App Designer components directly.

• Communication must go through the MATLAB HTML UI Data Channel:

  • app.HTMLComponent.Data (MATLAB → JS)

  • window.MATLAB.postMessage() (JS → MATLAB)

• MATLAB’s environment does not support Node.js runtime — only compiled/bundled JS.

• You cannot install NPM packages at runtime — only precompiled JS.

Performance notes:

• React will run fully client-side inside the HTML component sandbox.

• Fast enough for custom UIs, dashboards, toolbars, animations.

• Not fast enough for heavy 3D rendering (use Three.js instead).

In short:

✔ You can embed a React UI in MATLAB App Designer

✔ You can build a custom toolstrip using React

✖ You cannot integrate React with MATLAB's built-in UI controls

✔ You must communicate via the HTML-Component Data API

✔ 6. Can I design a custom toolstrip using HTML + JS + CSS?

Absolutely yes.
This is the recommended workflow for advanced UI:

Example:

html
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<div class="toolstrip">
  <button onclick="MATLAB.postMessage({cmd:'loadMesh'})">Load</button>
  <button onclick="MATLAB.postMessage({cmd:'computeEigen'})">Eigenbasis</button>
  <button onclick="MATLAB.postMessage({cmd:'filter'})">Filter</button>
</div>

Then in MATLAB:

matlab
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function HTMLComponentDataChanged(app, event)
    data = event.Data;
    switch data.cmd
        case 'loadMesh'
            loadMesh(app);
        case 'computeEigen'
            computeEigenbasis(app);
    end
end

This gives you a customizable toolstrip similar to:

• Signal Analyzer

• Classification Learner

• Wavelet Analyzer

Except built with modern HTML UI.

✔ 7. Should YOU use React? (My recommendation)

For your high-performance scientific app (bioctree GUI), the best choices are:

Best choice: HTML + CSS + lightweight JS (Vanilla / D3 / Svelte)

Because:

• Fastest load time

• Minimal overhead

• Easy communication with MATLAB

• Fully supports scientific UIs

Good choice: React, if you need advanced UI

Use React only if you need:

• complex state management

• many interactive panels

• collapsible toolstrips

• reusable UI components

Avoid: Angular, Vue, or big JS frameworks

They add heavy overhead and slow down App Designer.

✔ Summary

✔ If you want, I can provide:

Option A:

A fully working React-based toolstrip template for App Designer.

Option B:

A minimal HTML/JS/CSS toolstrip you can paste directly into your MATLAB HTML component.

Option C:

A blueprint for a MATLAB–JS bridge to send eigenbasis computation requests from HTML → MATLAB.

Which one would you like?

Short answer: YES — you can create a full HTML/CSS/JS component system living inside your MATLAB package (+bct/+ui) with classes like bct.ui.Toolstrip to automatically load HTML and manage two-way communication.

Below is the architecture, constraints, and a ready-to-use blueprint for building such a reusable UI library.

✔ 1. YES: MATLAB supports object-oriented HTML UI components

MATLAB App Designer + HTML UI component = a two-environment system:

MATLAB allows you to package these as classes:

diff
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+bct/+ui/Toolstrip.m
+bct/+ui/Sidebar.m
+bct/+ui/Console.m
+bct/+ui/PlotPanel.m
...
+bct/+ui/html/toolstrip.html
+bct/+ui/html/toolstrip.js
+bct/+ui/css/toolstrip.css

Every component can:

• load its own HTML

• inject JS variables

• receive JS → MATLAB messages

• expose MATLAB API methods

• behave like a class that App Designer uses

This is exactly how you should build your UI framework.

✔ 2. Architecture of a reusable MATLAB–HTML component class

Folder structure (recommended)

javascript
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+ bct
  + ui
    Toolstrip.m
    Sidebar.m
    Console.m
    Component.m        (base class)
    /html
        toolstrip.html
        sidebar.html
        console.html
    /js
        toolstrip.js
        sidebar.js
        console.js
    /css
        toolstrip.css
        sidebar.css
        console.css

✔ 3. Base component class (bct.ui.Component)

All custom UI components should inherit from a base class that holds:

• a handle to the HTML UI component

• a reference to the parent app

• automatic loading of HTML

• event routing from HTML → MATLAB

• methods for sending data to JS

Here is a real, working template:

matlab
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classdef Component < handle
    properties
        App          % reference to App Designer app
        HTML         % htmlUIComponent handle
        ID           % component ID for routing
    end

    methods
        function obj = Component(app, htmlComponent, id)
            obj.App = app;
            obj.HTML = htmlComponent;
            obj.ID = id;

            % Setup MATLAB->JS messaging
            obj.HTML.DataChangedFcn = @(src,evt) obj.onMessage(evt.Data);
        end

        function send(obj, message)
            % Send MATLAB → JS
            obj.HTML.Data = struct('id', obj.ID, 'message', message);
        end

        function onMessage(obj, data)
            % Overridden in subclasses
        end
    end
end

Now all your UI components can subclass this and implement onMessage.

✔ 4. Example: A Toolstrip component class

matlab
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classdef Toolstrip < bct.ui.Component
    
    methods
        function obj = Toolstrip(app, html)
            obj@bct.ui.Component(app, html, "Toolstrip");
            obj.loadHTML("toolstrip.html");
        end

        function loadHTML(obj, filename)
            fullPath = fullfile(fileparts(mfilename('fullpath')), ...
                                "html", filename);
            obj.HTML.HTMLSource = fullPath;
        end

        function onMessage(obj, data)
            cmd = data.cmd;

            switch cmd
                case "computeEigenbasis"
                    obj.App.computeEigenbasis();
                case "loadMesh"
                    obj.App.loadMesh();
                case "filter"
                    obj.App.openFilterUI();
            end
        end
    end
end

Now your toolstrip is a fully reusable, object-oriented UI component.

✔ 5. Example HTML for the Toolstrip (toolstrip.html)

html
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<link rel="stylesheet" href="../css/toolstrip.css">

<div id="bct-toolstrip">
    <button onclick="sendCmd('loadMesh')">Load Mesh</button>
    <button onclick="sendCmd('computeEigenbasis')">Compute Eigenbasis</button>
    <button onclick="sendCmd('filter')">Filters</button>
</div>

<script src="../js/toolstrip.js"></script>

✔ 6. toolstrip.js — communication JS → MATLAB

javascript
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function sendCmd(cmd) {
    window.MATLAB.postMessage({
        id: "Toolstrip",
        cmd: cmd
    });
}

This integrates with the MATLAB DataChangedFcn callback.

✔ 7. Using the Toolstrip inside your App Designer app

In your .mlapp file:

matlab
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properties
    Toolstrip  % instance of the UI component class
end

In startupFcn:

matlab
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app.Toolstrip = bct.ui.Toolstrip(app, app.HTMLToolstrip);

Where app.HTMLToolstrip is the HTML UI Component dropped into the app.

This line initializes:

• The Toolstrip class

• Loads toolstrip.html

• Binds all JS/MATLAB events

• Makes the toolstrip reusable across ANY app

✔ 8. Can MATLAB load CSS, Svelte, D3, or templates?

✔ CSS — fully supported

✔ D3 — fully supported

✔ Svelte — supported AFTER compilation to JS

You cannot run the Svelte compiler inside MATLAB, but you can:

bash
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npm run build

Then put the generated JavaScript bundle inside:

bash
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+bct/+ui/js/mycomponent.js

And include it in your HTML.

✔ 9. Can you build a reusable UI library like bct.ui?

YES.
MATLAB supports full packaging of object-oriented UI component classes.

Your architecture will look like this:

css
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bct.ui.Toolstrip  → loads HTML/JS/CSS → calls App backend
bct.ui.Sidebar    → loads HTML/JS/CSS → calls App backend
bct.ui.Console    → renders logs → receives JS requests
bct.ui.Viewport   → controls D3 canvas → visualizes mesh

Each component:

• Lives in its own HTML sandbox

• Communicates cleanly with MATLAB

• Can be dropped into any app

• Is versionable like a real UI framework

This is how professional MATLAB apps (e.g., Apps in MATLAB Online) are built internally.

✔ 10. Final Answer

YES

You can absolutely create a reusable UI component library inside +bct/+ui.

YES

You can implement classes like:

• bct.ui.Toolstrip

• bct.ui.Sidebar

• bct.ui.Console

• bct.ui.Panel

• bct.ui.Dock

YES

You can encapsulate MATLAB↔HTML communication inside each class.

YES

You can use vanilla JS, D3, or compiled Svelte/D3 tooling.

NO

You cannot reuse MATLAB’s internal toolstrip — you must build your own.

✔ Next Steps (choose one)

I can provide:

A. A ready-to-use bct.ui.Toolstrip full implementation

(HTML/CSS/JS/MATLAB class)

B. A skeleton for a full bct.ui framework

(styling, architecture, folder structure)

C. A sample app showing multiple custom HTML components working together

D. A modern UI design kit using Tailwind CSS + D3 inside MATLAB

Which one do you want?

Creating a modern UI design kit for MATLAB App Designer using Tailwind CSS + D3.js is 100% feasible and can completely transform the look and interactivity of your bct.ui system.

Below is a detailed, realistic breakdown of exactly what it requires, including:

• What is possible inside MATLAB’s HTML component

• Architectural requirements

• Build system (Tailwind)

• Runtime integration (JS + MATLAB messaging)

• Component packaging under +bct/+ui

• Limitations and workarounds

• A timeline and concrete deliverables

This is a real engineering blueprint.

⭐ OVERVIEW

A modern UI kit inside MATLAB would look like this:

• Beautiful Tailwind-styled HTML panels, buttons, tabs, toolbars

• D3.js interactive visualizations (custom sliders, knobs, graphs)

• Smooth animations, cards, shadows, dropdown menus

• A full custom toolstrip

• Modular reusable UI components

• MATLAB ↔ JS communication for interactivity

• No dependencies on MATLAB’s built-in controls

This allows you to build a professional-grade GUI similar to:

• VSCode panels

• ObservableHQ/D3 dashboards

• Web apps (React-like feel)

• Signal Analyzer–style views

⭐ WHAT THIS REQUIRES (High-Level)

To build this UI kit, you need:

✔ 1. A folder structure under +bct/+ui

Where you keep:

bash
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+ bct
  + ui
    Component.m
    Toolstrip.m
    Panel.m
    Sidebar.m
    Console.m
    Viewer.m
    /html/*.html
    /js/*.js
    /css/output.css        <-- Tailwind compiled file
    /tailwind-src/*.css    <-- Tailwind source

✔ 2. Embedding Tailwind in MATLAB’s HTML UI Component

Tailwind cannot run inside MATLAB.
But Tailwind’s compiled CSS works perfectly inside App Designer.

You must:

1. Write Tailwind source (Utility classes)

2. Run Tailwind CLI to produce output.css

3. Include the generated CSS file in your HTML:

html
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<link rel="stylesheet" href="../css/output.css">

This gives you:

• flex layouts

• spacing utilities

• grid

• typography

• shadows

• background gradients

• dark mode

• rounded corners

• responsive sizing

All inside MATLAB 🎉.

✔ 3. Embedding D3.js

Simple:

html
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<script src="https://cdn.jsdelivr.net/npm/d3@7"></script>

or include a local copy.

D3 works flawlessly in MATLAB’s HTML component, enabling:

• interactive SVG charts

• force-directed graphs

• cortical mesh overlays

• sliders, transitions, color scales

• HUD controls for visualization

✔ 4. Custom UI components as MATLAB classes

You create MATLAB classes such as:

mathematica
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bct.ui.Toolstrip
bct.ui.Sidebar
bct.ui.Panel
bct.ui.Console
bct.ui.D3Viewer
bct.ui.IconButton
bct.ui.Dropdown
bct.ui.Tabbar
bct.ui.Modal

Each class:

1. Loads its HTML file

2. Injects data/state

3. Listens for JS → MATLAB messages

4. Exposes MATLAB-side functionality

This gives you a reusable, consistent UI library.

✔ 5. Adding a MATLAB ↔ JavaScript bridge

You must wrap communication via:

MATLAB → JS

matlab
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app.HTMLComponent.Data = struct("cmd", "setMode", "value", 3);